In the context of seismic data processing, seismic migration refers to a process by which seismic events are re-located (in time or space) to the locations at which the events actually occurred rather than the location that the events were recorded at the surface, to thereby create a more accurate image of the subsurface being investigated. Demigration refers to a reverse of the migration process, which maps imaged data into modeled data for a given source/receiver configuration and given model parameters.
Demigration has developed into an important tool in for seismic imaging. Examples of demigration application include model building as described by H. Chauris and M. Benjemaa in their 2010 article entitled “Seismic Wave-Equation Demigration/Migration” published in volume 75, number 3, pages S111-S119 and incorporated herein by reference, multiple prediction as described by A. L. Pica, G. Poulain, B. David, M. Magesan, S. Baldock, T. Weisser, P Hugonnet, and P. Herrmann in their 2005 article entitled “3D Surface Related Multiple Modeling, Principles and Results” published in the 75th Annual International Meeting, SEG, Expanded abstracts, number 3, pages 2080-2083 (hereinafter “Pica”) and incorporated herein by reference, seismic inversion as described by S. Xu, D. Wang, F. Chen, G. Lambaré and Y. Zhang in their article entitled “Full Waveform Inversion for Reflected Seismic Data” published in the 74th Meeting, European Association of Geoscientists and Engineers, Expanded Abstracts and incorporated herein by reference and least-squares migration as described by T. Nemeth, C. Wu, and G. T. Schuster in their 1999 article entitled “Least-Squares Migration of Incomplete Reflection Data” published in Geophysics, volume 64, pages 208-221, incorporated herein by reference.
Demigration can be differentiated from modeling based on the fact that demigration uses reflectivity to predict seismic data while modeling uses velocity and density as input models. As described by N. Bleistein, J. K. Cohen and J. W. Stockwell in their 2001 reference entitled “Mathematics of Multidimensional Seismic Inversion” published by Springer Publishing Company and incorporated herein by reference, because migration is a technical “masterpiece” for reflectivity estimation, demigration is considered as the inverse or adjoint process of migration.
The origins of demigration can be traced to the concept of an “exploding reflector” which led to one-way wave equation based post-stack migration as described by D. Lowenthal, L. Lu, K. Roberson and J. Sherwood in their 1976 article entitled “The Wave Equation Applied to Migration” published in Geophysical Prospecting, volume 24, pages 380-399 and incorporated herein by reference. Looking forward from the Lowenthal et. al. reference, many migration methods have been developed and their corresponding demigration methods followed. More recently, reverse time migration (RTM) has become a standard migration tool and is now considered indispensable for delineating subtle hydrocarbon traps beneath complex overburden. In an effort to further improve imaging quality, Least-Squares Reverse Time Migration (LSRTM) as described by M. Wong, S. Ronen and B. Biondi in their 2011 article entitled “Least-Squares Reverse Time Migration/Inversion for Ocean Bottom Data: A Case Study” published in the 81st Annual International Meeting, SEG, Expanded Abstracts, pages 2369-2373, incorporated herein by reference, is emerging and requires several iterations of RTM and Reverse Time Demigration (RTDM).
Looking to the literature, One-Way Wave Equation Demigration (OWEDM) has been used to predict and suppress both surface related multiples as described by Pica and interbed multiples as described by A. Pica and L Delmas (hereinafter “Pica and Delmas”) in their 2008 article entitled “Wave Equation Based Internal Multiple Modeling in 3D” published in the 78th Annual International Meeting, SEG, Expanded Abstracts, pages 2476-2480 and incorporated herein by reference. Compared with the data driven demultiple techniques, e.g., Surface Related Multiple Elimination (SRME) as described by D. J. Verschuur and A. J. Berkhout in their 1997 article entitled “Estimation of Multiple Scattering by Iterative Inversion, Part II: Practical Aspects and Examples” published in Geophysics, Volume 62, pages 1596-1611, incorporated herein by reference, such model based multiple prediction methods are more general, better handle different acquisition geometries and are less sensitive to data sampling issues. However, when attempts are made to generalize the multiple prediction method from OWEDM to RTDM, it is found that a straightforward generalization leads to numerical instability.
Accordingly, it would be desirable to provide systems and methods that avoid the afore-described problems and drawbacks, and provide numerical computational stability when computing both surface related multiples and interbed multiples by a two-way propagation for improving the quality and capability of pre-stack depth imaging.